Any characteristic root \(\tau \)  of multiplicity \(r\ge 3\)  at \((t, x,\xi )\)  with \(t\ge 0\)  the point \((t, x,\tau ,\xi )\)  is a critical point, where we have \(F_p(t, x,\tau ,\xi ) = O\)  unless \(r=3\)  and \(t=0\) . Any double characteristic root \(\tau \)  at \((t, x,\xi )\)  with \(t\ge 0\)  the point \((t, x,\tau ,\xi )\)  is a critical point if \(t>0\)  while it is not necessarily critical point if \(t=0\)  ([1]}). Here is a simple example
\(P=(D_t^2-t^{\ell }D_x^2)(D_t+c\,D_x),\qquad \ell \in {\mathbb {N}},\quad x\in {\mathbb {R}},\;\;t\ge 0\) 
